SENSORLESS OPERATION OF FOC USING MODEL REFERENCE ADAPTIVE ...shodhganga.inflibnet.ac.in/bitstream/10603/93562/11/11_chapter5.pdf · The adaptive scheme for the MRAS estimator can

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Chapter 5

SENSORLESS OPERATION OF FOC USING

MODEL REFERENCE ADAPTIVE SYSTEM

WITH SLIDING MODE CONTROL

5.1. Introduction

The position or speed sensor (such as tachometer based speed

sensors, optical incremental sensors or electromechanical resolvers) in

induction motor drives impose many practical problems such as complexity

of hardware, difficulties in application to hostile environment, increased

cost, reduced reliability due to cables and sensors itself, difficulties of

mechanical attachment of sensor to the electric machine, increased axial

length of the machine and electromagnetic noise interference. To solve these

problems, various speed or position sensorless control schemes have been

developed for variable speed AC drives. In sensorless drives, no conventional

speed or position monitoring devices are used, instead the speed or/and

position signal is obtained by using monitored voltages and/or currents and

by utilizing mathematical models [81].

Although several schemes are available for sensorless operation of a

vector controlled drive, MRAS is popular because of its simplicity. SMC is

considered to be the appropriate methodology to replace PI controllers in

MRAS for the robust nonlinear control of induction motor drives due to its

order reduction, disturbance rejection, strong robustness and simple

implementation by means of power converter. SMC is a control strategy in

Variable Structure System (VSS) having a proper switching logic with high

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frequency discontinuous control actions depending on the system state,

disturbances and reference inputs.

In this chapter, based on the concepts of MRAS and SMC, a

mathematical model of speed estimation system is presented for sensorless

FOC induction motor drive. Based on this, simulation model of sensorless

FOC induction motor drive using MRAS-SMC is developed and validated

with extensive simulation results. Performance of the same is compared with

sensorless FOC induction motor drive model using MRAS-PI.

5.2. Model Reference Adaptive System

5.2.1. Overview of MRAS

The model reference approach takes advantage of using two

independent machine models, reference model and adjustable model for

estimating the same state variable. The estimation error between the

outputs of the two computational blocks is used to generate a proper

mechanism for adapting the speed. The difference between the two

estimated vectors is used to feed a PI controller. The output of the controller

is used to tune the adjustable model, which in turn actuates the rotor

speed. However, PI controllers may drop the performance level due to

the continuous variation in the machine parameters and operating

conditions in addition to nonlinearities contributed by the inverter.

5.2.2. MRAS Design: Hyper Stability Concepts

The adaption algorithm for MRAS can be taken into account the

overall stability of the system and to ensure that the estimated speed will

converge to the desired value with satisfactory dynamic characteristics [82].

The adaptive scheme for the MRAS estimator can be designed based on

Popov’s criteria for hyper stability concept [83], this relate to the stability

properties of a class of feedback systems. In general, a model reference

adaptive speed estimator system can be represented by an equivalent non-

linear feedback system which comprises a feed forward time-invariant linear

subsystem as well as a feedback non-linear time-varying subsystem. In Fig.

5.1, the input to the linear time-invariant system is u, which contains the

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stator voltage and currents and its output is y, which is the speed-tuning

signal. The output of the non-linear time invariant system is ω, and u = -ω.

The rotor speed estimation algorithm (adaptation mechanism) is chosen

according to Popov’s hyperstability theory, whereby the transfer function

matrix of the linear time invariant system must be strictly positive real and

the non-linear time-varying feedback system satisfies Popov’s integral

inequality. Fig. 5.1 shows the standard non linear time varying feedback

system, which is said to be asymptotically stable if it satisfies the following

two conditions:

i) The transfer function of the feed-forward linear time invariant block

must be strictly positive real.

ii) The error should converge asymptotically.

iii) The non linear time varying block satisfies the Popov’s integral

inequality:

2(0, )

0

tT

t y dtη ω γ= ≥ −∫

(5.1)

where y : Input vector

ω : Output vector of the feedback block

γ 2 : Finite positive constant

Fig. 5.1 Standard Non-Linear Time varying feedback system

5.3. MRAS Based Speed Estimation

MRAS computes a desired state called functional candidate using two

different models. In the rotor flux based MRAS, the rotor flux is used as an

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output value for the model to estimate the rotor speed. The MRAS scheme is

based on two independent machine models, the reference model and the

adaptive (adjustable) model for estimating the same state variable. Speed

estimation is done by comparing the output of the reference model with the

output of the adaptive model until the error between the two models

disappear. The reference model does not contain the speed to be computed,

which represents stator equation and is usually known as voltage model.

The reference value of the rotor flux components in the stationary frame are

generated from the monitored stator voltage and current components, which

are given by [1] and [34].

( )rr s s s s s

m

LV R i dt L i

Lα α α αψ σ = − − ∫

(5.2)

( )rr s s s s s

m

LV R i dt L i

Lβ β β βψ σ = − − ∫ (5.3)

0

0

r s ss sr

r s ss sm

V iR L pLp

V iR L pL

α α α

β β β

ψ σ

ψ σ

+ = − +

(5.4)

where,

11

(1 )(1 )s r

σσ σ

= −+ +

dp

dt=

The adaptive model contains the estimated rotor speed, which

represents the rotor equation and is usually known as the current model.

The adaptive values of rotor flux components are given by [1] and [34].

1ˆ ˆ ˆ ˆ( )r m s r r r r

r

L i dtα α α βψ ψ ω τ ψτ

= − −∫ (5.5)

1ˆ ˆ ˆ ˆ( )r m s r r r r

r

L i dtβ β β αψ ψ ω τ ψτ

= − +∫

(5.6)

1ˆ

ˆ ˆ

ˆ ˆ1ˆ

rr s rrm

r s rrr

r

iLp

i

α α α

β β β

ωψ ψτ

ψ ψτω

τ

− − = + −

(5.7)

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The instantaneous angular speed ωr of the rotor flux vector on an

open loop basis can be obtained from the measured voltages and currents.

The rotor flux vector angle and its derivative are expressed as:

1ˆ

tanˆ

r

r

β

α

ψφ

ψ−

=

(5.8)

( ) ( )2 2

ˆ ˆ ˆ ˆ

ˆ ˆ

r r r r

r r

p pp

α β β α

α β

ψ ψ ψ ψφ

ψ ψ

−=

+ (5.9)

Substituting (5.7) in (5.9),

2 2

ˆ ˆˆ

ˆ ˆ

r s r smr

r r r

i iLp

α β β α

α β

ψ ψφ ω

τ ψ ψ

−= +

+ (5.10)

The difference between the two estimated vectors is fed to an adaption

mechanism to generate estimated value of rotor speed which is used to tune

the adaptive model. The adaption mechanism of conventional rotor flux

MRAS is a simple fixed gain linear PI controller. The adaptive scheme for the

MRAS estimator can be designed based on Popov’s criteria for hyper stability

concept [82]. When the rotor flux of the adjustable model is in accordance

with that of the reference model the rotor speed of the adjustable model

becomes the real motor speed. The tuning signal, eω actuates the rotor

speed, which makes the error signal zero. The expression for estimated rotor

speed is given by [34]. MRAS of this kind are extensively used to identify

plant parameters and inaccessible variables. In designing the adaption

mechanism for a MRAS, it is important to consider the overall stability of the

system and to make sure that the estimated quantity will converge to the

desired value with suitable dynamic characteristics. Practical synthesis

technique for MRAS structures based on the concept of hyper stability was

described in [82]. Generally the models are linear time varying systems and

ωr is a variable, but for deriving an adaption mechanism, treat ωr as a

constant parameter of the reference model. State error equations are

obtained by subtracting (5.7) for the adjustable model from the

corresponding equations of the reference model shown in Fig. 5.2 as:

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Fig. 5.2 MRAS representation as a Non-Linear feedback system

[ ]

1

ˆˆ

ˆ 1

rr r rr

r rr rr

r

r

e ep

e e

α β α

β βα

ωψ τ

ω ωψ

ωτ

− − − = − + −

(5.11)

[ ] [ ][ ] [ ]e A e Iω ω ω= + −&

(5.12)

These equations describe a non linear feedback system since ὣr is a

function of the state error. In this hyper stability is assured provided that

the linear time invariant forward path transfer matrix is strictly positive real

and the non linear feedback satisfies Popov’s criterion for hyper stability.

5.3.1 Selection of Speed Adaption Law

For designing the adaptive laws the Popov’s inequality criterion is

made use of which consists of the following steps:

1. Transform the MRAS into an equivalent system called a non linear time

variable feedback system, which includes a feed forward linear model

and a non linear feedback system.

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2. Ensure strict positive realness of the feed forward linear model.

3. Design the adaptive laws which can ensure that the non linear

feedback block satisfies the Popov inequality given by (5.1).

4. Transfer the equivalent system back to the MRAS system.

Step 1: Transform the MRAS into an equivalent system

The first step to transform the MRAS into a Non Linear Time (NLT)

variable feedback system is shown in Fig. 5.2. Consider the system given by

(5.12) where the matrix A is constant and hence can be included in the

linear time invariant forward block. Now ὣr is a function of the state error

and is time varying in nature, hence along with the adaption mechanism

can be included in the non linear feedback system.

Step 2: Ensure strict positive realness of the feed forward linear

model

The second step is to ensure strict positive realness of the feed

forward linear model. For that consider a general system represented by,

x Ax Bu= +&

(5.13)

y Cx Du= +

(5.14)

Following Kalman-Yakubovich- Popov lemma (Positive Real Lemma),

Lemma 1: Let Z(s) =C (sI-A)-1B+D be a (p x p) transfer function matrix,

where A is Hurwitz, (A, B) is controllable, and (A, C) is observable. Then Z(s)

is strictly positive real if and only if there exist a positive definite symmetric

matrix P, matrices W and L, and a positive constant ԑ such that

T TPA A P P L Lε+ − = −

(5.15)

T TPB C L W= −

(5.16)

T TW W D D= +

(5.17)

So for the current system given by (5.11), it can be proved that A is

Hurwitz and also it is evident that (A, I) (as B = I) is controllable and (A, I) (as

C = I) is observable. Now for the system (5.16) can be rewritten as in (5.19),

where B = I, C = I and D = 0.

T TPA A P P L Lε+ − = −

(5.18)

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TP I=

(5.19)

Let ԑ=1 then

1 0

0 1P I

= =

(5.20)

Substituting the value of P from (5.20) and A from (5.11) in (5.18)

gives (5.21) which ensures that LTL is symmetrical positive definite and

hence Z(s) satisfies conditions of (5.18) and is strictly positive real.

2( 1) 0

20 ( 1)

rT T

r

PA A P P L Lτ

ε

τ

− + + − = − =

− +

(5.21)

Step 3: Ensure strict positive realness of the feed forward linear

model

The third step is to design the adaptive laws which can ensure that

the non linear feedback block satisfies the Popov inequality criterion given

by (5.1). For the system represented by Fig. 5.2, (5.1) can be written as:

2(0, )

0

tT

t e dtωη ω γ= ≥ −∫

for all t≥ 0 (5.22)

where γ2 is a positive constant,

eω is the output of the linear system,

ω is the output of the non linear system.

Substituting for eω and ‘ω’ in this inequality Popov’s criterion, the

present system becomes:

2

0

ˆ ˆ ˆ[ ][ ]t

r r r r r re e dtβ α α βψ ψ ω ω γ− − ≥ −∫

(5.23)

Let 1 2

0

ˆ ( ) ( )t

r f f dω τ τ τ= + ∫

(5.24)

then (5.23) becomes,

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21 2

0 0

ˆ ˆ[ ][ ( ( ) ( ) )t t

r r r r re e f f d dtβ α α βψ ψ ω τ τ τ γ− − + ≥ −∫ ∫

(5.25)

A solution to this inequality can be found through the relation as:

and

2 2 2

0

2

0

1 1. ( ) ( ) ( ( ) (0)) . (0)

2 2

( ) 0, 0

t

t

k f t f t dt f t f k f

f t dt k

= − ≥ −

≥ >

∫

∫

&

(5.26)

by assuming the values of f1(τ) and f2(τ) as given below and Popov’s

inequality criterion can be ensured by the following function as:

1ˆ ˆ ˆ ˆ( ) p r r r r p r r r rf K e e Kβ α α β β α α βτ ψ ψ ψ ψ ψ ψ = − = −

(5.27)

2ˆ ˆ ˆ ˆ( ) i r r r r i r r r rf K e e Kβ α α β β α α βτ ψ ψ ψ ψ ψ ψ = − = −

(5.28)

Hence the adaptive law for ὣr is given by PI adaptive law as:

0

ˆ ˆ ˆ ˆ ˆt

r p r r r r i r r r rK K dβ α α β β α α βω ψ ψ ψ ψ ψ ψ ψ ψ τ = − + − ∫

(5.29)

Step 4: Transfer equivalent system back to MRAS system

The last step is to transfer the equivalent system back to the MRAS

system whose representation is as shown in Fig. 5.3.

Fig. 5.3 Structure of MRAS based speed estimator

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ˆr p iK e K e dtω ωω = + ∫

(5.30)

It is important to design the adaptation mechanism of the MRAS

according to the hyper-stability concept, which will result in a stable and

quick response system where the convergence of the estimated value to the

actual value can be assured with suitable dynamic characteristics [1].

Popov’s criterion of hyper-stability for a globally asymptotically stable

system is used in deriving the speed estimation relation, which represents

the difference between the reference model and the adjustable model. The

parameter ε should be passed through a PI block that is found to be

satisfactory for the adaptive scheme, it gives the below objective function as:

ˆ ˆr r r reω β α α βψ ψ ψ ψ= −

(5.31)

The classical rotor flux MRAS speed observer, generally gives

reasonable speed estimation in the high and medium speed ranges.

However, at low speed regions, the performance deteriorates due to

integrator drift and initial condition problems and sensitivity to current

measurement noise [84].

Block diagram of MRAS based speed identification system is shown in

Fig. 5.4. The reference model is independent of of rotor speed. The

adjustable model requires stator currents and is dependent on the rotor

speed. The currents, is� and isβ and voltages, vs� and vsβ are taken from

motor terminals for the speed estimation.

Figure 5.5 describes the classical rotor flux MRAS-PI with both

references and adaptive models for rotor speed estimation. The parameter eω

should be passed through a PI block that is found to be satisfactory for the

adaptive scheme.

The speed estimated from MRAS is fed back to a speed controller in a

sensorless drive and is compared with the reference speed to get the

command output. Fig. 5.6 shows the block diagram of sensorless FOC

induction machine with MRAS-PI speed estimator.

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Fig. 5.4 Block diagram of MRAS based speed identification system

Fig. 5.5 Block diagram of MRAS-PI speed estimator

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Fig. 5.6 Block diagram of sensorless FOC with MRAS-PI speed estimator

5.3.2 Design of PI for MRAS Based Speed Estimator

The transfer function in (5.32) is obtained from (5.7) through

linearization with respect to a certain operating point [28]:

2

1( )2 2

1( )

1ˆ( )

r

rs

r rs

r

se

G

s

ω

ψτ

ω ω ωτ

+ ∆

= = ∆ − ∆ + +

(5.32)

Figure 5.7 depicts the block diagram of (5.32). Assuming ωs = 0 for

simplicity, damping factor ξ and natural angular frequency ωc can be

specified by using Kp and Ki as follows:

2

12 c

rp

r

K

ξωτ

ψ

−

= (5.33)

( )2

2

c

i

r

Kω

ψ= (5.34)

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In (5.33), Kp and Ki are fixed and calculated as a function of rotor flux

magnitude, ψr, in the steady state, which is obtained from current model.

Especially in transients and braking mode, it may cause to significant

errors. For the proposed method of work, rotor flux magnitude, ψr, which is

calculated from current model is replaced with that of voltage model. Since

voltage model have been selected as reference model and current model

must follow it, with this replacement, the proposed method will be in

prediction mode. On the other hand, Kp and Ki are on-line tuned with

respect to instantaneous variation of rotor flux, instead of its expected value

in steady state. In this way, rotor flux magnitude unexpected variations, will

be neutralized by on-line tuning of PI parameters and its resulting probable

instability will be prevented. This modification makes the proposed method

adaptive for various conditions. Prediction and adaptation are two

characteristics of the proposed method. This method does not complicate

the algorithm, because rotor flux magnitude, ψr, is continuously calculated

in the MRAS algorithm and thus there is no need for additional

computations.

Fig. 5.7 Speed estimator dynamics

5.4. Sliding Mode Control - Concepts

SMC is basically a non linear control method that alters the dynamics

of a non linear system by application of a high frequency switching control

varying system structure for stabilization. It is the motion of the system

trajectory along a chosen line, plane or surface of the state space. Control

structures are designed so that trajectories always move towards switching

condition and the ultimate trajectory will slide along the boundaries of the

control structures. Sliding mode can be reached in finite time due to

discontinuous control law. SMC is deterministic because only bounds of

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variations are considered and non linear since the corrective term is not

linear. It is robust since once on the sliding surface the system is robust to

bounded parameters variations and bounded disturbances. Sliding mode

control is an appropriate robust control method for the systems where

modeling inaccuracies, parameter variations and disturbances are present.

Sliding variables converge asymptotically and insensitive only against

matched perturbations and outputs with relative degree first are certain

disadvantages of sliding mode control. Chattering due to implementation

imperfections is the main weakness of SMC. The advantages of sliding mode

control comprise:

i. Low sensitivity to plant parameter uncertainty (bounded uncertainty)

ii. Reduced order modeling of plant dynamics

iii. Finite time convergence

Ability to result in very robust control system is the most important

aspect of the above scheme. In SM, a control law is designed so as to bring

the system trajectory on a predefined surface called the sliding surface.

Sliding phase is that phase where it slides along the predefined surface to

the equilibrium point and reaching phase is the phase before it touches the

sliding surface. A typical sliding mode control phase plot is shown in Fig 5.8,

where the equilibrium point is indicated by ‘O’ and reaching phase is

represented by the trajectory PQ and sliding phase by the trajectory QO. By

choosing appropriate control law, once the phase trajectory touches the

sliding surface, it is maintained in the sliding surface and finally converges

to the equilibrium point, the system dynamics will be governed by the

dynamics of the sliding surface. Then the system remains invariant to

parameter variations. In sliding mode controller, the system is controlled in

such a way that the error in the system states always move towards a

sliding surface. The sliding surface is defined with the tracking error of the

state and its rate of change as variables. The control input to the system is

decided by the distance of the error trajectory from the sliding surface and

its rate of convergence. At the intersection of the tracking error trajectory

with the sliding surface, the sign of the control input must be changed, thus

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the error trajectory is always forced to move towards the sliding surface. If

the desired error dynamics is:

2 1 0S X CX= + =

(5.35)

Then,

i) The control objective of sliding mode control is to reach S = 0 in finite

time.

ii) The manifold S = 0 is known as the sliding surface

iii) Once in the surface, the control must keep the trajectories “sliding" on

the surface: sliding mode

Fig. 5.8 Phases of Sliding mode control

5.5. Design of Sliding Mode Controller

The sliding mode controller design basically involves two steps:

1. Choose the sliding surface

2. Design a control law that brings the phase trajectory to the sliding

surface and maintains the trajectory on the sliding surface itself and

finally converge it to the equilibrium point.

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5.5.1. Selection of the Sliding Surface

Here only flux controller sliding surface is chosen and the respective

surface is taken as:

p iS K e K e dtψ ψ ψ ψ ψ= + ∫

(5.36)

ˆrefeψ ψ ψ= −

(5.37)

5.5.2. Selection of Control Law

For designing the control law, the induction machine state model in

stationary reference frame with stator current and stator flux as the state

variables is made use of. The induction machine state model in stationary

reference frame can be expressed as:

1s r r rs s r s s s s s

s r s r s s

R R Ri i i i V

L L L L L Lα α β α α β α

ωω ψ ψ

σ σ σ σ σ= − − − + + +&

(5.38)

1s r r rs s r s s s s s

s r s r s s

R R Ri i i i V

L L L L L Lβ β α β β α β

ωω ψ ψ

σ σ σ σ σ= − − − + + +&

(5.39)

where, 2

1 m

s r

L

L Lσ = −

s s s sV i Rα α αψ = −&

s s s sV i Rβ β βψ = −&

The control law for the sliding mode controllers has to be selected in

such a way that it should be stable. So, in order to prove the stability of

sliding mode controllers appropriate Lyapunov functions are chosen from

which the required discontinuous control law can be derived. Here choosing

a Lyapunov Function as:

21( )

2V S Sψ=

(5.40)

Taking the derivative of (5.40), it gives

( )V S S Sψ ψ= &&

(5.41)

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Taking the derivative of Sψ

ˆp ref i pS K K e Kψ ψ ψ ψ ψψ ψ= + − && &

(5.42)

Using (5.38) in (5.42) it can be written as:

1ˆ

pS M Kψ ψψ= − &&

(5.43)

where 1 1 1 2ˆ,p ref i p s sM K K e K F D V D Vψ ψ ψ ψ β αψ ψ= + = − + +&&

in which, ( )1ˆs

p s s s s

RF K i iψ α α β βψ ψ

ψ= +

1

ˆ

s

pD Kβ

ψ

ψ

ψ=

2ˆs

pD K αψ

ψ

ψ=

hence can be written as:

1 1 1 2s sS M F DV D Vψ β α= + − −&

(5.44)

which can be written as:

S M F Dν= + −& (5.45)

To ensure Lyapunov stability condition, choose the control law as follows:

1( ( ))v D M F Ks sat sα−= + + +

(5.46)

Substituting (5.46) in (5.45), (5.41) can be written as:

( )( )1( ) ( )V S S M F DD M F Ks sat sα−= + − + + +&

(5.47)

The above equation can be written as:

( )( )V S Ks sat sα= − +&

(5.48)

In this, the first derivative of the Lyapunov equation is negative

definite. Thus from (5.48) it is evident that, the sliding mode controller is

stable as it satisfies the Lyapunov’s stability condition.

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5.6. Sliding Mode MRAS Speed Observer

The sliding mode is a control technique to adjust feedback by

previously defining surface. The system which is to be controlled will be then

forced to that surface and the behaviour of the system slides to the desired

equilibrium point [84]. Induction machine as a nonlinear system can be

demonstrated as state space model in the canonical form [38]:

( , ) ( , ) ( )x A x t B x t u t= +&

(5.49)

where, [ ]min max, , , ( ( , )) , ,n n mx R A F u R rank B x t m u u u∈ ∈ ∈ = ∈

A time varying surface is defined in the state space by the scalar equation,

( ) 0S x =

(5.50)

The aim is to maintain the system motion on the manifold S(t), which is

defined as:

{ }( ) : ( , ) 0iS x x e x t= =

(5.51)

d

i i ie x x= −

(5.52)

where, e is state error or tracking error, xid is the desired state vector and xi

is the state vector. The control input u has to guarantee that the motion of

the system described in (5.49) is restricted to the manifold S in the state

space. The sliding mode control should be chosen such that the candidate

Lyapunov function, V which is a scalar function of S and its derivative

satisfies the Lyapunov stability criteria as:

21( ) ( )

2V S S x=

(5.53)

( ) ( ) ( )V S S x S x= &&

(5.54)

According to Lyapunov theory, if the time derivative of V(S) along a

system trajectory is negative definite, this will ensure that it constrains the

state trajectories to a point towards the sliding surface S(t) and once on the

surface, the system trajectories remain on the surface until the origin is

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reached asymptotically. Thus the sliding condition is achieved by the

following condition in (5.55) makes the surface an invariant set.

( ) ( )V x S xη≤ −&

(5.55)

where ‘η’ is a strictly positive constant on outside of S(t). Lyapunov stability

criteria will achieve as in (5.55) states that the squared “distance” to the

surface, measured by e(x)2, decreases along all system trajectories [38]. The

condition also implies that some disturbances or dynamic uncertainties can

be tolerated while keeping the surface an invariant.

Starting from any initial condition, the state trajectory reaches the

sliding surface in a finite time and then slides along the surface

exponentially with a time constant equal to 1/K. The control rule can be

written as:

( ) ( ) ( )eq swu t u t u t= + (5.56)

where, u(t) is the control vector, ueq(t) is the equivalent control vector

and usw(t) is the switching vector and must be calculated so that stability

condition as per (5.57) for the selected control is satisfied.

( ) ( ( , ))swu t sign S x tη=

(5.57)

where,

1

( ) 0

1

sign S

−

= = +

for

for

for

0

0

0

S

S

S

<

=

>

5.6.1. Construction of Sliding Mode Observer

The sliding mode control theory is now applied to the rotor flux MRAS

scheme for speed estimation by replacing the conventional constant gain PI

controller. With reference to dynamic model of induction machine and the

speed tuning signal, the time varying sliding surface S(t) is constructed as:

( , ) 0S x t e Ke dtω ω= + =∫ (5.58)

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where, K is the switching gain which is strictly positive constant. When the

system reaches the sliding surface, the error dynamics at the sliding

surface, S(t) = 0 will be forced to exponentially decay to zero. Thus,

0S e Keω ω= + =& &

(5.59)

e Keω ω= −&

(5.60)

substituting (5.59) in (5.54),

( ) ( )V S S e Keω ω= +& &

(5.61)

time derivative of (5.31) is

ˆ ˆ ˆ ˆ

r r r r r r r reω β α β α α β α βψ ψ ψ ψ ψ ψ ψ ψ= + − −& && &&

(5.62)

substituting the flux values

1 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ( )m m

r r r r s r r r r s r r r r r r r r

r r r r

L Le i i i

T T T Tω β α α β α β α α β β α β α β β α αψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ω ψ ψ ψ ψ= − + − − + − +& &&

(5.63)

ˆre a bω ω= −&

(5.64)

where,

1 1ˆ ˆ ˆ ˆm m

r r r r s r r r r s r r r

r r r r

L La i i i

T T T Tβ α α β α β α α β β α β αψ ψ ψ ψ ψ ψ ψ ψ ψ ψ= − + − − +& &

ˆ ˆr r r rb β β α αψ ψ ψ ψ= +

substituting in (5.61)

ˆ( ) ( )rV S S a Ke bω ω= + −&

(5.65)

the time derivative of V(S) is negative definite for the following conditions:

0

ˆ( ) 0

0

ra Ke bω ω

<

+ − =>

for

for

for

0

0

0

S

S

S

>

=

<

(5.66)

This can be attained when: ˆr eq swu uω = +

(5.67)

89

where, eq

a Keu

b

ω+=

. ( )swu sign Sη=

The block diagram for adaptive mechanism with sliding mode control

is presented in Fig. 5.9.

Fig. 5.9 Block diagram for adaptive mechanism with sliding mode control

The equivalent control, ueq defines the control action which keeps the

state trajectory on the sliding surface and the switching control, usw depends

on the sign of the switching surface and η is the hitting control gain which

makes (5.61) negative definite, whose main purpose is to make the sliding

condition viable and the value of η should be large enough to overcome the

effect of external disturbance [85], [86]. In the equivalent control, the

function b in the denominator may cause problems when its value

approaches zero. By allowing magnetization of the machine before starting

up and by adding a small positive value, this problem can be solved [87].

90

5.6.2. Chattering Problem

In order to account for the presence of modeling imprecision and of

disturbances, the control law has to be discontinuous across S(t). In

practice, switching is not instantaneous and value of S(t) is not known with

infinite precision. This leads to oscillations in the state vector at finite

frequency, referred to as chattering as shown in Fig. 5.10. It results in high

heat losses in electrical power circuits, high wear of moving mechanical

parts and low control accuracy [38], [39]. The controller given will have

chattering near sliding surface due to the presence of sign function. These

drastic changes of input can be avoided by introducing a boundary layer

with width, φ [85]. By replacing sign(s) with sat(S/φ), it becomes,

( / )swu sat Sη φ=

(5.68)

where,

( / )( / )

( / )

sign Ssat S

S

φφ

φ

=

( / ) 1

( / ) 1

if S

if S

φ

φ

≥

<

a) Ideal b) Practical (with chattering)

Fig. 5.10 Sliding mode

A natural solution to reduce the chattering in the estimated speed is

by means of a Low-Pass Filter (LPF) as in (5.69).

' 1

1sw swu u

sµ=

+ (5.69)

91

The LPF also eliminates the spikes due to the differentiation of the

fluxes in the estimated speed. In this work, LPF is used to reduce the

chattering in the estimated rotor speed.

5.7. Implementation of MRAS-SM in FOC Drive

Simulation model for sensorless FOC induction motor drive using

MRAS-PI is developed based on the block diagram shown in Fig. 5.6. The

induction motor model, induction motor controller, MRAS-SM speed

estimator and the PWM inverter are shown in Fig. 5.11. For control

purposes, the motor model is first transformed into the d-q axis model. The

machine variables can be transformed into d-q components by using the �-β

to d-q transformation. The block calculates the voltages applied in the d-q

axis. The d-q axis voltages are converted into the �-β components. Both of

these transformations require the rotor flux angle, which is calculated as

shown in the block diagram. A non-linear adaption mechanism is proposed

to replace the fixed gain PI controller which is conventionally used for rotor

flux MRAS observer. The scheme is based on SM theory, where speed

estimation adaption law is derived based on Lyapunov theory to ensure

Fig. 5.11 Block diagram of FOC induction motor with MRAS-SM

92

estimation stability with fast error dynamics. To evaluate the performance of

the MRAS with SM scheme in sensorless FOC induction motor drive,

simulation model has been developed using MATLAB/Simulink as shown in

Fig. 5.11. The induction motor is fed by a SVM voltage source inverter

having parameters as mentioned in Appendix-I and the rotor speed is

estimated by MRAS-SM speed observer. The values of sliding mode

parameters for the simulation are: K = 100 and η = 0.1 which are obtained

by trial and error.

5.8. Simulation Results and Discussion

In the simulation, for comparing the performance of torque, the motor

starts from a standstill state with reference speed 140 rad/sec and

application of a load torque, TL = 5 Nm at time, t = 1 sec. Figures 5 .12 and

5.13 show the performance of electromagnetic torque with time for cases

with MRAS-PI and MRAS-SM respectively. The motor torque has a high

initial value in the speed acceleration zone, then the value decreases to zero

and increases to the applied load torque in both cases. In both cases

chattering appears in torque response, which will be filtered out by the

mechanical system inertia due to high frequency changes in the torque.

Fig. 5.12 Torque response for MRAS-PI

0

2.5

5

7.5

10

12.5

15

0.0 0.3 0.5 0.8 1.0 1.3 1.5 1.8 2.0Time (sec)

Torq

ue

(Nm

)

ωref =140rad/sec

93

Fig. 5.13 Torque Response for MRAS-SM

Figure 5.14 shows the speed response for start-up and steady state of

the induction machine for three reference speeds, 50 rad/sec, 100 rad/sec

and 140 rad/sec for sensorless FOC drive using MRAS-PI. Figure 5.15

shows the speed responses for three reference speeds, 50 rad/sec, 100

rad/sec and 140 rad/sec for sensorless FOC drive using MRAS-SM. The

applied load torque, TL = 5 Nm at t = 1 sec for all the reference speeds for

both cases. In both the cases, the estimated speed is close to the actual

speed and tracking the reference in all the three speed ranges. MRAS-SM

shows better performance when reference speed is 50 rad/sec, compared to

the case where reference speed is 150 rad/sec. Thus MRAS-SM can provide

better performance in low speed range compared to MRAS-PI.

In Fig. 5.15, for case-2, chattering, a problem associated with sliding

mode control is more when reference speed is 140 rad/sec compared to 50

rad/sec. However, when a load torque is applied, the chattering is more or

less same in all speed ranges.

0.0

2.5

5.0

7.5

10.0

12.5

15.0

0.0 0.3 0.5 0.8 1.0 1.3 1.5 1.8 2.0

Time (sec)

Torq

ue

(Nm

)ωref =140rad/sec

Fig

Fig.

0

20

40

60

80

100

120

140

160

0.0 0.2

Ro

tor

spee

d (

rad

/sec

)

0

20

40

60

80

100

120

140

160

0.0 0.2

Ro

tor

spee

d (

rad

/sec

)R

oto

r sp

eed

(ra

d/s

ec)

94

Fig.5.14 Rotor speed vs. Time using MRAS

Fig. 5.15 Rotor speed vs. Time using MRAS

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Estimated speed

Actual speed

Reference speed

Time (sec)

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Estimated speedActual speedReference speed

Time (sec)Time (sec)

ime using MRAS-PI

ime using MRAS-SM

1.8 2.0

Estimated speed

Actual speed

Reference speed

1.8 2.0

Estimated speedActual speedReference speed

95

The error in rotor speed for all the speed ranges is shown in Fig. 5.16.

From the figures, it is clear that MRAS-SM scheme provides accurate speed

estimation in high, medium and low speed ranges.

a) for ωref = 50 rad/sec

b) for ωref = 100 rad/sec

c) for ωref = 140 rad/sec

Fig. 5.16 Speed error for MRAS-SM

-10

-5

0

5

10

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Err

or(

rad

/sec

)

Time (sec)

-10

-5

0

5

10

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Err

or(

rad

/sec

)

Time (sec)

-10

-5

0

5

10

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Err

or(

rad

/sec

)

Time (sec)

96

Variation of current in stator windings for the three reference speeds are

shown in Fig. 5.17. As in the case of electromagnetic torque, the current

signals in stator windings have high values in transient state. In the steady

state, the motor torque only has to compensate the friction and the load

torque.

a) for ωref = 50 rad/sec

b) for ωref = 100 rad/sec

c) for ωref = 140 rad/sec

Fig. 5.17 Stator current for MRAS-SM

-8.0

-4.0

0.0

4.0

8.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Sta

tor

curr

ent(

A)

Time (sec)

-8.0

-4.0

0.0

4.0

8.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Sta

tor

curr

entr

(A)

Time (sec)

-8.0

-4.0

0.0

4.0

8.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Sta

tor

curr

ent

(A)

Time (sec)

97

Figure 5.18 presents the estimated rectangular components of the

rotor flux in the loaded condition in time range between 1.80 sec and 2.0sec

for all the three speed ranges.

a) for ωref = 50 rad/sec

b) for ωref = 100 rad/sec

c) for ωref = 140 rad/sec

Fig. 5.18 Estimated rotor flux for MRAS-SM

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

1.80 1.85 1.90 1.95 2.00

Roto

r fl

ux (

Wb

)

Time (sec)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

1.80 1.85 1.90 1.95 2.00

Roto

r fl

ux (

Wb

)

Time (sec)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

1.80 1.85 1.90 1.95 2.00

Roto

r fl

ux (

Wb

)

Time (sec)

98

The speed tuning signal which is the input to the sliding mode

observer for various reference speeds in all the speed ranges are shown in

Fig. 5.19.

a) for ωref = 50 rad/sec

b) for ωref = 100 rad/sec

c) for ωref = 140 rad/sec

Fig. 5.19 Speed tuning signal of MRAS-SM

-0.10

-0.05

0.00

0.05

0.10

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

e ω

Time (sec)

-0.10

-0.05

0.00

0.05

0.10

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

e ω

Time (sec)

-0.10

-0.05

0.00

0.05

0.10

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

e ω

Time (sec)

99

Figure 5.20 shows the circular locus of rotor flux vector when the

reference speed is 100 rad/sec. The estimated rotor flux angle for the

reference speed is 50 rad/sec is shown in Fig. 5.21.

Fig. 5.20 Estimated rotor flux locus for MRAS-SM

Fig. 5.21 Estimated flux angle for MRAS-SM

By comparing and observing all the simulation results, it ensures that

the proposed MRAS with sliding mode observer works well when the

parameters are precisely measured and do not change during operation.

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

ψrd(W

b)

ψrd

(Wb)

-2

-1

0

1

2

1.80 1.85 1.90 1.95 2.00

Flu

x a

ngle

(ra

d)

Time (sec)

100

5.10. Summary

In this chapter, a novel adaption mechanism using sliding mode

control is proposed to replace the classical PI controller in MRAS observer

for SVM fed induction machine. The adaption mechanism is based on

Lyapunov theory to ensure stability with fast error dynamics. The chattering

problem of the estimated speed is effectively reduced by using a low pass

filter. The performance of the developed MRAS sliding mode observer has

been illustrated and the speed and torque responses are compared with that

of PI controller by simulation results. The developed adaption mechanism

shows better performance in a wide range of speed compared with that of PI

controller, for which performance deteriorates at low speed. The speed

estimation by the rotor flux MRAS with sliding mode observer has ensured

very good accuracy in both transient and steady state for all ranges of speed

control.

5.11. Publications Related to this Chapter

International Conference: 1. G. K. Nisha, Z. V. Lakaparampil and S. Ushakumari, “FFT Analysis for Field

Oriented Control of SPWM and SVPWM Inverter fed Induction Machine With and

Without Sensor”, International conference on Advance Engineering

Technology(ICAET’13), Mysore, India, pp. 34-43, 27 January 2013.

2. G. K. Nisha, Z. V. Lakaparampil and S. Ushakumari, “Sensorless Vector Control of

SVPWM Inverter fed Induction Machine using Sliding Mode”, IEEE International

conference on Green Technology (ICGT’12), Thiruvananthapuram, India, pp. 29-36,

18-20 December 2012.

International Journal: 1. G. K. Nisha, Z. V. Lakaparampil and S. Ushakumari, “Performance Study of Field

Oriented Controlled Induction Machine in Field Weakening using SPWM and SVM

fed Inverters”, International Review of Modeling and Simulations, vol. 6, no. 3, pp.

741-752, June 2013.

2. G. K. Nisha, Z. V. Lakaparampil and S. Ushakumari, “FFT Analysis for Field

Oriented Control of SPWM and SVPWM Inverter fed Induction Machine With and

Without Sensor”, International journal of Advanced Electrical Engineering, vol. 2, no.

4, pp. 151-160, June 2013.